MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  flimclslem Structured version   Visualization version   GIF version

Theorem flimclslem 22081
Description: Lemma for flimcls 22082. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
flimcls.2 𝐹 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
Assertion
Ref Expression
flimclslem ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝑆𝐹𝐴 ∈ (𝐽 fLim 𝐹)))

Proof of Theorem flimclslem
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 flimcls.2 . . 3 𝐹 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
2 topontop 21011 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
323ad2ant1 1163 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ Top)
4 eqid 2765 . . . . . . . . 9 𝐽 = 𝐽
54neisspw 21205 . . . . . . . 8 (𝐽 ∈ Top → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 𝐽)
63, 5syl 17 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 𝐽)
7 toponuni 21012 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
873ad2ant1 1163 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑋 = 𝐽)
98pweqd 4322 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝒫 𝑋 = 𝒫 𝐽)
106, 9sseqtr4d 3804 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝒫 𝑋)
11 toponmax 21024 . . . . . . . . . 10 (𝐽 ∈ (TopOn‘𝑋) → 𝑋𝐽)
12 elpw2g 4987 . . . . . . . . . 10 (𝑋𝐽 → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1311, 12syl 17 . . . . . . . . 9 (𝐽 ∈ (TopOn‘𝑋) → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1413biimpar 469 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋) → 𝑆 ∈ 𝒫 𝑋)
15143adant3 1162 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ 𝒫 𝑋)
1615snssd 4496 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝑆} ⊆ 𝒫 𝑋)
1710, 16unssd 3953 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ 𝒫 𝑋)
18 ssun2 3941 . . . . . 6 {𝑆} ⊆ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆})
19113ad2ant1 1163 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑋𝐽)
20 simp2 1167 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆𝑋)
2119, 20ssexd 4968 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ V)
22 snnzg 4464 . . . . . . 7 (𝑆 ∈ V → {𝑆} ≠ ∅)
2321, 22syl 17 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝑆} ≠ ∅)
24 ssn0 4140 . . . . . 6 (({𝑆} ⊆ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ∧ {𝑆} ≠ ∅) → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ≠ ∅)
2518, 23, 24sylancr 581 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ≠ ∅)
2620, 8sseqtrd 3803 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 𝐽)
27 simp3 1168 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐴 ∈ ((cls‘𝐽)‘𝑆))
284neindisj 21215 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ (𝐴 ∈ ((cls‘𝐽)‘𝑆) ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝐴}))) → (𝑥𝑆) ≠ ∅)
2928expr 448 . . . . . . . . . . 11 (((𝐽 ∈ Top ∧ 𝑆 𝐽) ∧ 𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝑥 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑥𝑆) ≠ ∅))
303, 26, 27, 29syl21anc 866 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝑥 ∈ ((nei‘𝐽)‘{𝐴}) → (𝑥𝑆) ≠ ∅))
3130imp 395 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑥𝑆) ≠ ∅)
32 elsni 4353 . . . . . . . . . . 11 (𝑦 ∈ {𝑆} → 𝑦 = 𝑆)
3332ineq2d 3978 . . . . . . . . . 10 (𝑦 ∈ {𝑆} → (𝑥𝑦) = (𝑥𝑆))
3433neeq1d 2996 . . . . . . . . 9 (𝑦 ∈ {𝑆} → ((𝑥𝑦) ≠ ∅ ↔ (𝑥𝑆) ≠ ∅))
3531, 34syl5ibrcom 238 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝐴})) → (𝑦 ∈ {𝑆} → (𝑥𝑦) ≠ ∅))
3635ralrimiv 3112 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) ∧ 𝑥 ∈ ((nei‘𝐽)‘{𝐴})) → ∀𝑦 ∈ {𝑆} (𝑥𝑦) ≠ ∅)
3736ralrimiva 3113 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ∀𝑥 ∈ ((nei‘𝐽)‘{𝐴})∀𝑦 ∈ {𝑆} (𝑥𝑦) ≠ ∅)
38 simp1 1166 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐽 ∈ (TopOn‘𝑋))
394clsss3 21157 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑆 𝐽) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
403, 26, 39syl2anc 579 . . . . . . . . . . . 12 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((cls‘𝐽)‘𝑆) ⊆ 𝐽)
4140, 27sseldd 3764 . . . . . . . . . . 11 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐴 𝐽)
4241, 8eleqtrrd 2847 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐴𝑋)
4342snssd 4496 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝐴} ⊆ 𝑋)
44 snnzg 4464 . . . . . . . . . 10 (𝐴 ∈ ((cls‘𝐽)‘𝑆) → {𝐴} ≠ ∅)
45443ad2ant3 1165 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝐴} ≠ ∅)
46 neifil 21977 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝐴} ⊆ 𝑋 ∧ {𝐴} ≠ ∅) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
4738, 43, 45, 46syl3anc 1490 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋))
48 filfbas 21945 . . . . . . . 8 (((nei‘𝐽)‘{𝐴}) ∈ (Fil‘𝑋) → ((nei‘𝐽)‘{𝐴}) ∈ (fBas‘𝑋))
4947, 48syl 17 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ∈ (fBas‘𝑋))
50 ne0i 4087 . . . . . . . . . . 11 (𝐴 ∈ ((cls‘𝐽)‘𝑆) → ((cls‘𝐽)‘𝑆) ≠ ∅)
51503ad2ant3 1165 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((cls‘𝐽)‘𝑆) ≠ ∅)
52 cls0 21178 . . . . . . . . . . 11 (𝐽 ∈ Top → ((cls‘𝐽)‘∅) = ∅)
533, 52syl 17 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((cls‘𝐽)‘∅) = ∅)
5451, 53neeqtrrd 3011 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((cls‘𝐽)‘𝑆) ≠ ((cls‘𝐽)‘∅))
55 fveq2 6379 . . . . . . . . . 10 (𝑆 = ∅ → ((cls‘𝐽)‘𝑆) = ((cls‘𝐽)‘∅))
5655necon3i 2969 . . . . . . . . 9 (((cls‘𝐽)‘𝑆) ≠ ((cls‘𝐽)‘∅) → 𝑆 ≠ ∅)
5754, 56syl 17 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ≠ ∅)
58 snfbas 21963 . . . . . . . 8 ((𝑆𝑋𝑆 ≠ ∅ ∧ 𝑋𝐽) → {𝑆} ∈ (fBas‘𝑋))
5920, 57, 19, 58syl3anc 1490 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → {𝑆} ∈ (fBas‘𝑋))
60 fbunfip 21966 . . . . . . 7 ((((nei‘𝐽)‘{𝐴}) ∈ (fBas‘𝑋) ∧ {𝑆} ∈ (fBas‘𝑋)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ↔ ∀𝑥 ∈ ((nei‘𝐽)‘{𝐴})∀𝑦 ∈ {𝑆} (𝑥𝑦) ≠ ∅))
6149, 59, 60syl2anc 579 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ↔ ∀𝑥 ∈ ((nei‘𝐽)‘{𝐴})∀𝑦 ∈ {𝑆} (𝑥𝑦) ≠ ∅))
6237, 61mpbird 248 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
63 fsubbas 21964 . . . . . 6 (𝑋𝐽 → ((fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋) ↔ ((((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))))
6419, 63syl 17 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋) ↔ ((((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ≠ ∅ ∧ ¬ ∅ ∈ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))))
6517, 25, 62, 64mpbir3and 1442 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋))
66 fgcl 21975 . . . 4 ((fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋) → (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))) ∈ (Fil‘𝑋))
6765, 66syl 17 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))) ∈ (Fil‘𝑋))
681, 67syl5eqel 2848 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐹 ∈ (Fil‘𝑋))
69 fvex 6392 . . . . . 6 ((nei‘𝐽)‘{𝐴}) ∈ V
70 snex 5066 . . . . . 6 {𝑆} ∈ V
7169, 70unex 7158 . . . . 5 (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ∈ V
72 ssfii 8536 . . . . 5 ((((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ∈ V → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
7371, 72ax-mp 5 . . . 4 (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))
74 ssfg 21969 . . . . . 6 ((fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ∈ (fBas‘𝑋) → (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))))
7565, 74syl 17 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))))
7675, 1syl6sseqr 3814 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (fi‘(((nei‘𝐽)‘{𝐴}) ∪ {𝑆})) ⊆ 𝐹)
7773, 76syl5ss 3774 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ⊆ 𝐹)
78 snssg 4471 . . . . 5 (𝑆 ∈ V → (𝑆 ∈ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ↔ {𝑆} ⊆ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
7921, 78syl 17 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝑆 ∈ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}) ↔ {𝑆} ⊆ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆})))
8018, 79mpbiri 249 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆 ∈ (((nei‘𝐽)‘{𝐴}) ∪ {𝑆}))
8177, 80sseldd 3764 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝑆𝐹)
8277unssad 3954 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)
83 elflim 22068 . . . 4 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
8438, 68, 83syl2anc 579 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝐴 ∈ (𝐽 fLim 𝐹) ↔ (𝐴𝑋 ∧ ((nei‘𝐽)‘{𝐴}) ⊆ 𝐹)))
8542, 82, 84mpbir2and 704 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → 𝐴 ∈ (𝐽 fLim 𝐹))
8668, 81, 853jca 1158 1 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑆𝑋𝐴 ∈ ((cls‘𝐽)‘𝑆)) → (𝐹 ∈ (Fil‘𝑋) ∧ 𝑆𝐹𝐴 ∈ (𝐽 fLim 𝐹)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wcel 2155  wne 2937  wral 3055  Vcvv 3350  cun 3732  cin 3733  wss 3734  c0 4081  𝒫 cpw 4317  {csn 4336   cuni 4596  cfv 6070  (class class class)co 6846  ficfi 8527  fBascfbas 20021  filGencfg 20022  Topctop 20991  TopOnctopon 21008  clsccl 21116  neicnei 21195  Filcfil 21942   fLim cflim 22031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-iin 4681  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-ov 6849  df-oprab 6850  df-mpt2 6851  df-om 7268  df-wrecs 7614  df-recs 7676  df-rdg 7714  df-1o 7768  df-oadd 7772  df-er 7951  df-en 8165  df-fin 8168  df-fi 8528  df-fbas 20030  df-fg 20031  df-top 20992  df-topon 21009  df-cld 21117  df-ntr 21118  df-cls 21119  df-nei 21196  df-fil 21943  df-flim 22036
This theorem is referenced by:  flimcls  22082
  Copyright terms: Public domain W3C validator